Tài liệu Fourier and Spectral Applications part 6 doc

558 Chapter 13. Fourier and Spectral Applications

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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)

for (j=2;j<=m;j++) {

j2=j+j;

p[j] += (SQR(w1[j2])+SQR(w1[j2-1])

+SQR(w1[m44-j2])+SQR(w1[m43-j2]));

}

den += sumw;

}

den *= m4; Correct normalization.

for (j=1;j<=m;j++) p[j] /= den; Normalize the output.

free_vector(w2,1,m);

free_vector(w1,1,m4);

}

CITED REFERENCES AND FURTHER READING:

Oppenheim, A.V., and Schafer, R.W. 1989, Discrete-Time Signal Processing (Englewood Cliffs,

NJ: Prentice-Hall). [1]

Harris, F.J. 1978, Proceedings of the IEEE, vol. 66, pp. 51–83. [2]

Childers, D.G. (ed.) 1978, Modern Spectrum Analysis (New York: IEEE Press), paper by P.D.

Welch. [3]

Champeney, D.C. 1973, Fourier Transforms and Their Physical Applications (New York: Aca￾demic Press).

Elliott, D.F., and Rao, K.R. 1982, Fast Transforms: Algorithms, Analyses, Applications (New

York: Academic Press).

Bloomfield, P. 1976, Fourier Analysis of Time Series – An Introduction (New York: Wiley).

Rabiner, L.R., and Gold, B. 1975, Theory and Application of Digital Signal Processing(Englewood

Cliffs, NJ: Prentice-Hall).

13.5 Digital Filtering in the Time Domain

Suppose that you have a signal that you want to filter digitally. For example, perhaps

you want to apply high-pass or low-pass filtering, to eliminate noise at low or high frequencies

respectively; or perhaps the interesting part of your signal lies only in a certain frequency

band, so that you need a bandpass filter. Or, if your measurements are contaminated by 60

Hz power-line interference, you may need a notch filter to remove only a narrow band around

that frequency. This section speaks particularly about the case in which you have chosen to

do such filtering in the time domain.

Before continuing, we hope you will reconsider this choice. Remember how convenient

it is to filter in the Fourier domain. You just take your whole data record, FFT it, multiply

the FFT output by a filter function H(f), and then do an inverse FFT to get back a filtered

data set in time domain. Here is some additional background on the Fourier technique that

you will want to take into account.

• Remember that you must define your filter function H(f) for both positive and

negative frequencies, and that the magnitude of the frequency extremes is always

the Nyquist frequency 1/(2∆), where ∆ is the sampling interval. The magnitude

of the smallest nonzero frequencies in the FFT is ±1/(N∆), where N is the

number of (complex) points in the FFT. The positive and negative frequencies to

which this filter are applied are arranged in wrap-around order.

• If the measured data are real, and you want the filtered output also to be real, then

your arbitrary filter function should obey H(−f) = H(f)*. You can arrange this

most easily by picking an H that is real and even in f.